AHP's, HAP's and DAP's: How Potassium Currents Regulate the ...

Journal of Computational Neuroscience 15, 367–389, 2003c© 2003 Kluwer Academic Publishers. Manufactured in The Netherlands.

AHP’s, HAP’s and DAP’s: How Potassium Currents Regulatethe Excitability of Rat Supraoptic Neurones

PETER ROPERNational Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health,

Bethesda, MD 20892, [email protected]

JOSEPH CALLAWAY, TALENT SHEVCHENKO, RYOICHI TERUYAMA AND WILLIAM ARMSTRONGDepartment of Anatomy and Neurobiology, University of Tennessee College of Medicine, 855 Monroe Avenue,

Memphis, TN 38163, USA

Received March 10, 2003; Revised August 20, 2003; Accepted August 22, 2003

Action Editor: John Rinzel

Abstract. We have constructed mathematical models of the electrical activity of two hypothalamic supraopticneuro-secretory cell-types, and we support our models with new calcium imaging and in vitro electrophysiologicaldata. These cells are neurones that project to the pituitary gland and secrete either of two hormones, oxytocinor vasopressin, into the blood from their axonal terminals. Oxytocin-secreting and vasopressin-secreting cells areclosely related and physically they differ only subtly, however when physiologically stressed their discharge patternsare dramatically distinct. We first show how each potassium current contributes to the action-potentials and after-potentials observed in these cells, and we show how these after-potentials are correlated to intra-cellular calciumelevations. We then show how these currents regulate the excitability of these cells and consequently shape theirdischarge pattern.

1. Introduction

Oxytocin (OT) and arginine vasopressin (AVP) magno-cellular neurosecretory cells (MNC’s) in the rat hy-pothalamus are the best described peptidergic neuronesin the mammalian brain. They project to the neurohy-pophysis, and when electrically stimulated their termi-nals secrete either OT or AVP directly into the blood viaexocytosis. Release is primarily a response to: osmoticstress (AVP and OT); haemorrhage (AVP and OT); par-turition (OT); and suckling (OT), and is a function ofboth firing rate and of the discharge pattern. OT andAVP cells are morphologically and electrically verysimilar and their action potentials cannot easily be dis-tinguished. Spikes from both cell types are sterotypic

with a fast rise and a repolarization that briefly over-shoots the rest potential.

After spiking the membrane potential evinces sev-eral after-potentials before relaxing to the rest poten-tial. First the repolarizing overshoot merges into anevanescent hyperpolarized after-potential (HAP1) (seeFig. 1a), which has a slower decay than the passive re-sponse of the membrane and so is actively sustained bya conductance (Andrew and Dudek, 1984b; Bourqueet al., 1985). A substantial fraction of MNC’s then ex-press a slow depolarizing overshoot that follows the ini-tial HAP (Andrew and Dudek, 1983) (see Fig. 1c). Thisdepolarizing after-potential (DAP) is both calcium- andvoltage-dependent (Bourque, 1986; Andrew, 1987; Liet al., 1995), and is more frequently expressed in AVP

368 Roper et al.

Figure 1. Spike after-potentials in SON MNC’s: (a) The hyperpolarized after-potential (HAP) follows each spike and typically hyperpolarizesthe cell by ∼ 7.5 mV and lasts for 25–125 ms. (b) An after-hyperpolarizing potential (AHP) follows each spike train. It decays mono-exponentiallywith τ ∼ 500 ms and has a maximum amplitude ∼12.5 mV. (c) The depolarized after-potential (DAP) is preferentially expressed in AVP cells.DAP’s can last for several seconds and a single DAP can depolarize the cell by ∼3 mV. Note that the spike has been truncated for clarity. Thetraces are taken from whole cell (a, b) and sharp electrode (c) recordings in the slice and explant respectively.

cells and less so in OT MNC’s (Armstrong et al., 1994).However, it is labile and so its presence or absence doesnot discriminate cell type.

Trains of spikes also activate a long-lasting hyperpo-larization of the membrane potential, the AHP shownin Fig. 1b (Andrew and Dudek, 1984b; Bourque et al.,1985; Armstrong et al., 1994; Kirkpatrick and Bourque,1996). The AHP decays with a single time constant(400–500 ms) (Armstrong et al., 1994; Teruyama andArmstrong, 2002), is abolished by removal of calcium,and is markedly attenuated by apamin (Bourque andBrown, 1987; Armstrong et al., 1994; Kirkpatrick andBourque, 1996; Kirkpatrick 1997).

We have constructed a single-compartment mathe-matical model, based on a set of Hodgkin-Huxley equa-tions, that contains the minimum number of currentsand the simplest calcium dynamics needed to repro-duce in vitro MNC electrical activity. We explore inparticular how the different potassium currents sculpt

these after-potentials and show how they contribute tothe regulation of excitability in these cells. We alsopresent data from simultaneous calcium fluorescenceand electrical recordings to show how electrical activitycorrelates with calcium elevation and how spike after-potentials relate to intra-cellular calcium transients. Wefirst reconstruct the evoked action potentials and after-potentials shown by a generic SON MNC and we com-pare the model with cellular activity. We then extendour model to include features specific to first AVP andthen to OT cells, and we finally show how these featurescontribute to aspects of the cells’ discharge pattern.

2. Methods

2.1. Electrophysiology

We have borrowed from both sharp electrode and wholecell recordings made in the lab to demonstrate some of

AHP’s, HAP’s and DAP’s 369

the properties of SON neurones tested in the model.All of the examples used to illustrate these prop-erties have been described in the literature by our-selves and others. Sharp electrode recordings weremade from the hypothalamo-neurohypophysial explantpreparation (Stern and Armstrong, 1996; Teruyama andArmstrong, 2002). Briefly, virgin adult female rats weredeeply anesthetized with sodium pentobarbital (50mg/kg, i.p.) and perfused through the heart with coldmedium in which NaCl was replaced by an equiosmo-lar amount of sucrose. A ventral hypothalamic explantwas removed with iris scissors and placed in an incuba-tion chamber. The incubation medium consisted of (inmM): 25 NaHCO3, 3 KCl, 1.24 NaH2PO4, 124 NaCl,10 glucose, 2 CaCl2, 1.3 MgCl2, and 0.2 ascorbicacid. The medium was saturated with 95% O2− 5%CO2, with a pH of 7.3–7.4 and an osmolality of 290–300 mOsm/kg H2O; it was warmed to 33–34◦C. Intra-cellular recording and labeling, signal digitization, anddata analysis were made as previously described usingsharp electrodes filled with 1.5 M potassium acetate,0.1 M potassium chloride.

Whole cell recordings were made from 300 µm hy-pothalamic slices under visual guidance, using proce-dures similar to those reported by Stern et al. (1999)and Wilson and Callaway (2000). These animals werealso perfused under anesthesia (see above) with ice-cold medium before slicing on a vibrating microtome(Campden Inst. VSL, or Leica VT1000S). Pipetteswere filled with solution consisting of (in mM): K-gluconate or K-methylsulfate 140, NaCl 4, KCl 8.2,MgCl 0.6, HEPES 10, Mg-ATP 4, Na-GTP 0.3, andEGTA (0.2) (pH 7.4). Recordings were made with ei-ther a Neurodata IR283 intracellular recording bridgeamplifier or an Axopatch 200B amplifier (Axon In-struments). The effects of 100 nM Iberiotoxin (IbTX)(Sigma/RBI and Latoxon) on spikes were evaluatedwith a paired t-test. All values in the text are mean± s.d.

2.2. Calcium Imaging

We used calcium imaging to obtain values for the am-plitude and decay of calcium transients during spik-ing. The procedures were similar to those describedin Wilson and Callaway (2000). Horizontal slices(300 µm) through the SON of adult, virgin femalerat brains were made after anesthesia and perfusionas described above. The extracellular solution con-sisted of (in mM): 25 NaHCO3, 3 KCl, 1.24 NaH2PO4,124 NaCl, 10 glucose, 2 CaCl2, 1.3 MgCl2, and 0.2

ascorbic acid. Slices were imaged on an OlympusBX50WI microscope using a 40× water immersiblelens (0.8 n.a.). The pipette filling solution consisted of(in mM): K-gluconate or K-methylsulfate 135, NaCl 4,KCl 4, HEPES 10, Mg-ATP 1, Na-GTP 1.0, and fura-2(Na salt) 0.1 (pH 7.4). Recordings were made with aNeurodata IR283 intracellular recording bridge ampli-fier. High speed fura-2 fluorescence images were ob-tained using the Imago Sensicam (T.I.L.L. Photonics,Planegg, Germany), with a 12 bit, 640×480 chip (read-out rate = 12.5 MHz; scale = 0.246 µm/pixel with the40× objective). For excitation, light of either 340 or380 nm was provided by a Polychrome II monochro-mometer (T.I.L.L. Photonics) controlled with analogvoltage commands and using a Hammamatsu L2841-01 75W Mercury-Xenon arc lamp. The frame rate was40–50 Hz, and pixels were typically binned (4 × 4) toincrease the signal to noise ratio. Software for data ac-quisition and analysis is based on that developed anddescribed by Lasser-Ross et al. (1991) and modified byJ.C. Callaway, and allowed simultaneous recording ofelectrical and fluorescence signals from a Computer-boards 16 bit A-D board.

Changes in bulk calcium ([Ca2+]i ) were estimatedby measuring the change in fluorescence intensity at380 nm (�F) divided by a baseline fluorescence (F) toobtain the fractional change in [Ca2+]i , �F/F . Base-line fluorescence was first corrected for tissue autoflu-orescence by subtracting the background fluorescencefrom a region near the filled cell. The value �F/F wasfurther corrected for bleaching during an exposure bysubtracting time matched, filtered (3 Hz) control curvesof F at a hyperpolarized holding potential (<−70 mV),where no calcium entry could be detected.

To obtain information about the range of change inabsolute [Ca2+]i , we first measured resting [Ca2+]i inseveral neurones near −70 mV by the ratioing method(Grynkiewicz et al., 1985) using the following formula:

[Ca2+]i = KdR − Rmin

Rmax − R

F380max

F380min(1)

where R is the 340/380 nm intensity ratio in the neuroncorrected for background fluorescence at each wave-length, Rmin and Rmax are the minimum and max-imum ratios from a calibration done on this set-upusing Ca2+ -EGTA solutions and fura-2 provided byMolecular Probes and diluted in our K gluconatepatch solution. The calibration produced a Kd forfura-2 of 226 nM, Rmin of 0.416, Rmax of 10.37 andF380max/F380min of 9.32. The change in Ca2+ to a

370 Roper et al.

stimulus was then calculated from �F/F using thefollowing formula, derived by Wilson and Callaway(2000):

[Ca2+]i

= −�FF Kd + [Ca2+]rest

((�FF − 1

) F380maxF380min

+ 1)

1Kd

[Ca2+]rest�FF

F380maxF380min

+ (�FF − 1 + F380max

F380min

)(2)

where [Ca2+]rest is the resting level measured in thatneuron. This formula did not require obtaining a max-imal fluorescence change (i.e. by calcium loading). Inthe text all experimental results will be cited as mean± s.d.

2.3. Mathematical Model

2.3.1. Electrical Activity. The relative distribution ofcurrents over the cell is unknown and so as a first ap-proximation we model the cell as a single (electrical)point, and neglect spatial effects. While there is evi-dence that SON dendrites do contribute to electrotonus(Armstrong and Smith, 1990) and are excitable (Bainsand Ferguson, 1999), this simplified approach is suffi-cient to explain the gross aspects of the cell’s dischargeand allows us to study interactions between currents.We model electrical activity as a Hodgkin-Huxley-typesystem:

dV

dt= − 1

C (INa + ICa + IK + IA + Ic

+ IAHP + ISOR + Ileak) (3)

where INa represents the fast sodium current; ICa a highthreshold calcium current; IK and IA denote the delayedrectifier and the A-current respectively; Ic and IAHP twocalcium-dependent potassium currents; ISOR is the sus-tained outward-rectifier (Stern and Armstrong, 1995);and Ileak is a leak current that accounts for the mem-brane’s passive decay to rest.

All voltage- and calcium-dependent currents havethe standard activation/inactivation form:

Iγ (t) = gγ mα(t) hβ(t) (V − Erev) or

Iγ (t) = gγ mα(t) (V − Erev) (4)

Where the first equation describes an inactivating, andthe second a non-inactivating current, and each currentIγ has conductance gγ and reversal potential Erev.

Any activation (e.g. m(t)) or inactivation (h(t)) func-tion x(t) evolves to its equilibrium state x∞ with timeconstant τx , according to

d

dtx(t) = x∞ − x(t)

τx(5)

Wherever possible we have drawn on published val-ues (e.g. Cobbett et al., 1989; Nagatomo et al., 1995)for activation functions and their time constants, how-ever we have had to adjust many of these parame-ters to build a consistent model. These adjustmentshave typically been of the order of a <10 mV shift ofthe midpoint of the activation/inactivation curve, andcould be accounted for by differences in experimen-tal protocols (see also Borg-Graham, 1999; Golowaschet al., 2002). The resulting activation and inactiva-tion functions, and their corresponding time constants,are given in the appendix. We have assumed a spe-cific capacitance, C = τ/R, of 1 µF cm−2 (Hille,2001), and fitted conductances (measured in mS cm−2)accordingly.

2.3.2. Electrical Currents

2.3.2.1. Potassium Currents. MNC’s express a de-layed rectifier, IK (Cobbett et al., 1989); a tran-sient ‘A’-type current IA (Bourque, 1988; Cobbettet al., 1989; Nagatomo et al., 1995); an AHP cur-rent, IAHP (Andrew and Dudek, 1984b; Bourque etal., 1985) which is carried by SK channels (Bourqueand Brown, 1987; Kirkpatrick and Bourque, 1995,1996; Kirkpatrick, 1997); and a voltage- and calcium-dependent potassium current, Ic, (Dopico et al., 1999)which is carried by BK channels. OT-cells, butnot VP-cells, also possess a slowly-activating, time-and voltage-dependent, non-inactivating, sustained-outward-rectification (ISOR) (Stern and Armstrong,1995).

2.3.2.2. Inward Currents. The ubiquitous fast sodiumcurrent, INa, (Cobbett and Mason, 1987) mediates theupstroke of the action potential. INa has the usual cu-bic activation and linear inactivation, and parametershave been adjusted to give activity consistent withexperiment. We have assumed that INa activates in-stantaneously, so that m(t) = m∞, and that the cur-rent reverses at ENa = +50 mV. SON MNC’s expressseveral high- and intermediate-voltage activated cal-cium channels (Fisher and Bourque, 1995; Foehringand Armstrong, 1996; Luther and Tasker, 2000; Joux


et al., 2001), but, in rat, low-threshold currents areinconsistently observed (but see Fisher and Bourque,1995). As a consequence of their depolarized activa-tions, these calcium currents are only active during anaction potential and we therefore approximate the fam-ily of Ca2+ currents by a single, high-voltage activated,non-inactivating calcium current ICa.

2.3.2.3. Leak Currents. The leak current models allprocesses and pumps that comprise the passive re-sponse of the membrane. It accounts for the membranetime constant (12–16 ms; Armstrong et al., 1994 andhas a reversal potential of Eleak = −65 mV, and con-ductance G leak = 0.083 mS cm−2

Ileak = G leak(V − Eleak) (6)

A significant contribution to Ileak is a persistent potas-sium current, IK ,leak, that is always active when the cellis at rest (Han et al., 2003). In AVP cells IK ,leak appearsto have a component that can be modulated by changesin intra-cellular calcium, and this calcium-dependencyis also voltage-dependent (Li and Hatton, 1997b). Toexamine the effects of calcium and voltage on the leakcurrent we will later decompose Ileak into a linear com-bination of a sodium leak current and a potassium leakcurrent.

Ileak ≡ G leak(V − Vrest) = INa,leak + IK ,leak (7)

2.3.3. Calcium Concentrations. Our data (see later)suggests that intra-cellular calcium is compartmental-ized, possibly by buffering processes. There appearto be (at least) three distinct compartments, each ofwhich activates a single calcium dependent process—Ic, IAHP or IK ,leak. Rather than investigating a com-plete spatial model for calcium diffusion in these cells,we model this compartmentalization with three distinctpools of calcium (see also Borg-Graham, 1987). Thesepools correspond to the local concentrations in threegeographically disparate regions, as shown in Fig. 2.Each is homogeneous and for simplicity we assumethat there is no diffusion between them. For exam-ple, the calcium sensor for a BK channel is locatedwithin 1 µm of the calcium channel (Storm, 1993),and senses the rapidly fluctuating calcium concentra-tion in a small domain about the channel (Van Goor etal., 2001). In contrast, the potassium component to theleak current appears to be modulated by the mean so-matic (bulk) calcium concentration, [Ca2+]i , (see later)

Figure 2. Calcium domains and preferential activation of potas-sium channels. Ellipses denote the proposed locations of the calciumreceptor for the respective channel. We propose that: BK receptorsare co-localized with their channel; SK channels are attached to thecell membrane, but distal from the channel; IK ,leak receptors are in-ternal to the cell, and so sense bulk and not micro-domain calciumconcentrations. (i) Dashed curve indicates putative calcium domainassociated with Ic . The BK channel is close to the calcium channeland so experiences a high concentration, rapidly attenuating calciumtransient. (ii) Dotted curve denotes domain associated with IAHP. SKchannels are further from the Ca2+ channel and so are affected by asmaller, slower (τ ∼ 500 ms) calcium signal. (iii) The receptors forIK ,leak are still further from the site of Ca2+ influx and so the cur-rent is modulated by bulk calcium rather than some local transient.An example of the relative amplitudes and time-scales of these threetransients is shown in Fig. 3.

and does not seem to be affected by localized calciumtransients.

Calcium concentrations evolve according to (Plant,1978)

d

dtCγ = −αγ ICa − 1

τγ

(Cγ − Cr ) (8)

The subscript γ labels the pool of calcium, so thatγ = {BK, SK, i} depending on whether the pool acti-vates Ic, IAHP or IK ,leak. Cγ denotes the concentrationof calcium in that pool; αγ relates the amplitude of thecalcium current to calcium influx, and also reflects thesize of each pool; the time constant τγ approximatesthe buffering and diffusion processes that return the cal-cium transient to rest; and Cr (=113 nM) denotes thebasal calcium concentration. The relative evolutions of

372 Roper et al.

Figure 3. Putative amplitudes and time-scales of model calcium transients (recall Fig. 2) in response to three evoked action potentials. Allconcentrations are in nM.

each compartment are compared in Fig. 3, and of thethree evolution equations will be discussed individuallyin the text.

This model ignores the fine details of calcium home-ostatis, but our intent here is to model and understandthe role and dynamics of potassium currents, and thedegree to which they are affected by [Ca2+]i . Calciumthus plays a passive role in our analysis, and so wehave used a phenomenological model that most simplyreproduces its trajectory.

2.4. Numerical Methods

Simulations were undertaken with the differen-tial equation solver XPPaut (Ermentrout, 2002)(downloadable from http://www.math.pitt.edu/∼bard/xpp/

xpp.html), and several different integration algorithms(e.g. backwards Euler, Dormund-Prince and CVode)and time steps (most frequently δt = 0.017 ms) wereused to check accuracy and stability.

3. Results

3.1. Calcium Concentrations

Calcium is associated with the activation of both BKand SK potassium channels and has also been impli-cated as a key modulator of the resting potassium leakcurrent, IK ,leak (Li and Hatton, 1997b). Our simulta-neous electrical recording and calcium imaging showsthat intracellular calcium concentrations in MNC’s in-crease only when the cell is electrically active, andthat there is little sub-threshold calcium entry. Thelargest contribution to calcium increase is probablyCa2+ influx through voltage gated channels, but thereis also a component due to release from internal stores(Lambert et al., 1994; Li and Hatton, 1997a). Howeverour experimental protocol cannot distinguish betweenthese two sources. Clearance after activity is mediatedby extrusion via pumps and exchangers, and uptake intomitochondria and intracellular stores (Komori et al.,2001).


The mean basal calcium concentration in the soma is113 ± 60 nM (n = 45), and a single spike transientlyraises this by 19.8 ± 6.3 nM (n = 11), which thenclears mono-exponentially with a time constant of τ =0.99 ± 0.5 s, (n = 11).

Clearance after trains of spikes is more complex andwe used two different protocols to examine the conse-quent transients. First, to evoke DAP’s that were rela-tively free from contamination by the AHP, we used a3 spike train at 20 Hz, and varied the membrane po-tential with continuous current to achieve the largestDAP that was subthreshold to spiking (Armstrong etal., 1994) (left panel Fig. 4a). The calcium transientsfrom individual spikes did not sum linearly, and thecalcium increase following 3 spikes at 20 Hz was43.1 ± 13.2 nM (n = 15). In contrast to single spikes,most calcium transients elicited by triplets of spikesat 20 Hz showed a bi-exponential decay with a fast(mean time-constant τ f = 0.33 ± 0.14 s, n = 11/15)and a slow (τs = 1.94 ± 0.95 s, n = 15) compo-nent (middle panel Fig. 4a). This suggests that thereare two dominant mechanisms contributing to clear-ance, for example extrusion close to the membranecould provide the faster mechanism, and release and re-uptake into intracellular stores (Li and Hatton, 1997a)the slower one. Alternatively these two mechanismscould be the Na+ -Ca2+ exchanger and the calciumpump. The presence of a slow time constant for cal-cium decay depends only on the spiking history andnot on presence or absence of the DAP. For example,when a subset of the cells were slightly hyperpolarizedto inhibit the DAP, three spikes still evoked calciumtransients that decayed with two time constants in 5/7neurons, and these were not significantly different fromthose associated with a DAP at more depolarized po-tentials: 0.28 ± 0.03 s (n = 5/7) and 2.28 ± 1.3 s(n = 7/7).

We further used trains of 20 spikes at 10 and 20 Hzin order to evoke AHPs (Kirkpatrick and Bourque,1996) and examined the associated calcium transients(Fig. 4b). Although some neurons revealed two timeconstants in this protocol, the faster τ was smalland inconsistent even within a neuron, so we esti-mated the decay from a single exponential. In con-trast to single spikes or short (3 spike) trains, the timeconstants after these longer trains had a leftwardlyskewed distribution, and the mean (2.78 ± 1.1 s) wasdiscrepant from the median. Given this skew, it isworthwhile considering the median value, which was2.33 s.

The median AHP time constant, 656 ms, that wedetermined in this protocol is close to those thatwe have reported previously (400–500 ms) in nor-mal rats with sharp electrode recordings (Armstronget al., 1994; Stern and Armstrong, 1996; Teruyamaand Armstrong, 2002), and from Fig. 4 it is clearthat while the clearance of bulk calcium is trackedby the time-course of the DAP, it is dissociated fromthe decay of the AHP. A similar dissociation has beennoted in CA1 pyramidal cells (Jahromi et al., 1999),(but cf. Lancaster and Zucker, 1994; Lasser-Ross etal., 1997). We propose that in MNCs this anomaly iscaused by cytosolic Ca2+ compartmentalization (Chadand Eckert, 1984; Simon and Llinas, 1985; Van Gooret al., 2001), and that the calcium sensor for each AHPchannel is distant from site of influx but close to thecell membrane (see e.g. Yuen and Durand, 1991; Sah1992). It therefore experiences a global calcium con-centration, but can be rapidly affected by membranepumps and diffusion and so the domain around it de-cays more quickly than does the bulk calcium. Al-ternatively it is possible the AHP/[Ca2+]i dissociationrepresents a morphological segregation such that theAHP channels are confined to the dendrites andexperience a faster transient due to higher surfaceto volume ratios (cf. Wilson and Callaway, 2000).However, while we have found that dendritic cal-cium transients do generally decay faster (data notshown), they rarely match the decay of the AHP.In addition our preparation (300 µm slice) typicallyincludes a limited dendritic contribution, and further-more even fully deafferented cells still possess a sub-stantial AHP (see e.g. Fig. 8 of Oliet and Bourque,1992).

We treat the bulk calcium and the calcium domainsurrounding each AHP channel as being two distinct,and disconnected, compartments (see Fig. 2) whosecalcium concentrations are denoted Ci and CSK respec-tively, and which both evolve according to Eq. (8). Wehave adjusted the pre-factor αi to fit the rise amplitude(19.8 nM per spike), and the pre-factor αSK to fit theactivation of the AHP. We have set the decay time-constant for the AHP compartment to be equal to themedian AHP decay time-constant, i.e. τSK = 656 ms,and to further simplify the model we have used asingle, slow, time-constant for bulk calcium decay,τi = 2.33 s. The evolution of each compartment inresponse to a short spike train is shown in Fig. 3, andis compared with that of CBK, the micro-domain sur-rounding each BK channel. Note that as the volume

374 Roper et al.

Figure 4. (a) Left Panel: An average of 15 DAP’s from different neurons evoked with 3 spikes at 20 Hz. The baseline membrane potential wasoffset to be near −50 mV for all cells to reduce DAP amplitude variability and obtain a smooth decay. The decay was fit to a single exponentialof 1.851 s. Middle Panel: The average somatic calcium response for the 15 DAP’s. Each response was normalized to its peak. This curve wasfit with a double exponential with a fast time constant of 0.165 s and a slow time constant of 1.683 s. The vertical lines show the region of thiscurve (i.e. 1 ≤ t ≤ 6 s) matching the DAP decay shown in the left panel. Right Panel: The decay of the averaged DAP is plotted against theaverage Ca2+ decay. For calcium, values were converted to nM using the average peak value (41 nM) in order to appreciate the relationshipbetween elevations in bulk calcium and the activation of the DAP. The linear fit to these responses is shown. (b) Calcium rise and clearance fortrains of evoked spikes at 10 Hz and 20 Hz. Clearance is mono-exponential and independent of frequency (τ ∼ 2.4 s). Data taken from wholecell recordings from the SON of a hypothalamic slice.

of the domain decreases (recall Fig. 2), the maximumamplitude of the transient increases but decays morequickly, as one should expect from surface-to-volumeconsiderations.

3.2. IA and IK : Spike Repolarization and Overshoot

The transient outward current, IA, (Bourque, 1988;Cobbett et al., 1989; Nagatomo et al., 1995) has a


Figure 5. Activation and inactivation of the A-current. The cur-rent is significantly inactivated at rest, and completely inactivatedby a depolarization of ∼10 mV. Inactivation can be removed by aconditioning hyperpolarization.

quartic activation and a linear inactivation (i.e. α = 4,β = 1 for Eq. (4)), while the delayed rectifier, IK ,is a non-inactivating current with a cubic activation(Cobbett et al., 1989). IA is significantly inactivated atrest (−65 mV) and almost completely inactivated bya depolarization to −60 mV (see Fig. 5). In MNCs,IA is blocked by 4-aminopyridine (4-AP), but not bytetraethylamonium (TEA), while the opposite is truefor IK , and so these currents can be distinguished bytheir pharmacology.

Figure 6. Both IK and IA contribute to spike repolarization, and pharmacological block of either current significantly broadens the actionpotential (Bourque, 1988; Bourque and Renaud, 1988). The effects of TEA and 4-AP are reproduced in the model by (a) setting G K = 0 and(b) setting G A = 0 (cf. Fig. 12 of Bourque, 1988).

Spikes evoked from rest are significantly broadenedby the application of either 1 mM 4-AP (Bourque,1988) or TEA (Bourque and Renaud, 1988) but theirwidths are unaffected by 10 nM charybdotoxin (Dopicoet al., 1999), implying that both IA and IK but not Ic

mediate spike repolarization. These protocols can besimulated in the model by setting either G A or G K tobe zero, and the resulting effect on spike width is shownin Fig. 6. The relative contribution of each current torepolarization depends upon the degree of inactivationof IA, and hence the holding potential, and so spikesevoked from a depolarized holding potential will bebroader than those evoked from a more hyperpolarizedone. The HAP depends on IA (Bourque, 1988; Bourqueet al., 1998), and in the model this is caused by IA over-shooting the rest potential. However the HAP also hasa component that depends on Ic and this will be exam-ined later.

3.2.1. IA and Spike Latency. A conditioning hyper-polarization to <−75 mV removes inactivation of IA

(Fig. 5) and delays the occurrence of the first spikewhen the cell is subsequently depolarized (Bourque,1988; Fisher et al., 1998; Fisher and Bourque, 1998)(see Fig. 7), and this latency is abolished by 6 mM4-AP (Bourque, 1988; Stern and Armstrong, 1997).The rapid activation of IA appears as a knee on thefast rising phase of the depolarization, and develops

376 Roper et al.

Figure 7. Inactivation of A-current and latency to spiking—(a) ex-periment, (b) model. Cell given a conditional hyperpolarization toremove inactivation of IA . Note initial rapid rise of membrane poten-tial, followed by notch due to activation of IA and subsequent slowdepolarization as IA inactivates. Experimental trace from a wholecell recording from the SON in a hypothalamic slice.

into a notch as the magnitude of the hyperpolarizationincreases. The slower rise of the potential after the ini-tial rapid activation reflects A-current inactivation andthe consequent removal of the hyperpolarizing current.This protocol is successfully reproduced by the model(Fig. 7).

3.3. IAHP: Spike Frequency Adaptation and the AHP

The AHP following trains of spikes has a maximumamplitude ∼12.5 mV, decays with a time constant of400–500 ms, and is activated only by calcium andnot voltage (Bourque and Brown, 1987; Armstronget al., 1994; Kirkpatrick, 1996; Kirkpatrick, 1997) (seeFig. 1b). Associated with AHP activation during a spiketrain is a progressive increase in the inter-spike inter-val, which is termed spike frequency adaptation (SFA).Both SFA and the AHP are abolished by removal of cal-cium, and markedly attenuated by apamin (Bourqueand Brown, 1987; Armstrong et al., 1994; Kirkpatrick

and Bourque, 1996; Kirkpatrick, 1997), and so bothare caused by the activation of calcium-dependent SKpotassium channels (Vergara et al., 1998). A thirdcalcium-dependent after-hyperpolarization, slower andweaker than the AHP, has been described (Greffrathet al., 1998) but will not be included here.

Our simultaneous voltage and calcium measure-ments suggest that the AHP is progressively activatedby increasing calcium, (see Fig. 4b), however the cal-cium dependence of IAHP has not been quantifiedfor MNC’s. In other preparations it is described as anon-inactivating current, with a quadratic activation,q(CSK), that follows a Boltzman function (Yuen andDurand, 1991; Aradi and Holmes, 1999), such that

q∞(CSK) =(

1 + exp

[−1.120

− 2.508 log

(CSK − Cr

1000

)])−1

(9)

where CSK denotes the concentration in the AHP cal-cium domain, and Cr (=113 nM) the resting calciumconcentration. We assume that the current activates in-stantaneously (i.e. q ≡ q∞) since calcium dynamicsare already slow, and so

IAHP = GAHP q2∞(V − EK ) (10)

Once activity has ceased, the AHP decays sharply,typically within 30 ms of cessation of the spike train(Greffrath et al., 1998). Thus q does not saturate, orreach a plateau, during normal activity. If calcium wereto saturate q, the AHP would not decay immediately,but would instead initially show a lengthy, flat hyper-polarization which would persist until calcium decayedbeyond the point of saturation.

The decay of the AHP can be fit to a single exponen-tial for the majority of AVP cells and roughly half of allOT cells (Teruyama and Armstrong, 2002). An expres-sion for decay may be derived by applying the chainrule to Eq. (9), so that q = CSK dq/dCSK. If calciumdecay is also mono-exponential (recall that τs dom-inates clearance) then dq/dCSK must remain constant,and so q(CSK) must be approximately linear for most ofthe decay. Thus the activation of IAHP must be confinedto the (approximately) linear portion of the activationcurve, and so 0 < q2 < 0.35 (see Fig. 8), implying thatCSK must be bounded, possibly by buffering and re-uptake mechanisms. A significant fraction of OT cellsshow a bi-exponential AHP decay, and this variation


Figure 8. Activation of IAHP as a function of calcium concentration,CSK. Bold curve shows steady-state activation q2∞ (see Eq. (9)), anddashed line (q) is a linear fit to q2∞. The model indicates first thatthe AHP does not saturate, and second that for mono-exponentialdecay of the AHP, q2 must be confined to the linear region, thusq2

max < 0.35.

could be explained if q has a higher upper bound forOT cells and so departs further from linearity.

CSK evolves according to Eq. (8), and has decay con-stant τSK = 656 ms. We have chosen the step-size,αSK = 1.6, to match the observed progression of spike-frequency adaptation during a spike-train.

The conductance GAHP is now determined by theamplitude of current needed to hyperpolarize the cellby the maximum AHP amplitude when q2 = 0.35.For MNC’s in normal animals, the mean maximumAHP across cell types is ∼12.5 mV, (Armstrong et al.,1994; Stern and Armstrong, 1996), and so GAHP =0.18 mS cm−2.

Comparisons between experimental data and themodel are shown in Fig. 9 both for the AHP and forspike frequency adaptation.

3.4. Ic and the HAP

The fast, calcium- and voltage-dependent K+ current,Ic, is carried by BK channels, whose steady-state acti-vation, p∞(CBK, V ), can be factored into the product ofa Hill- and a Boltzman-function (Dopico et al., 1999):

p∞ =[

1 + 470

C2.38BK

]−1 [1 + exp

×((−V − 140 log10 CBK + 370

)7.4

) ]−1

(11)

where CBK denotes the calcium concentration in thepool sensed by the BK channel. Dopico et al. (1999)

determined mean open- and close-times for a singlesomatic BK channel, and so an activation time constantfor Ic may be derived from their data (Hille, 2001), suchthat τp = 1.22 ms .

Block of BK channels with 100 nM iberiotoxin(IbTX) shows that Ic is significantly activated dur-ing a single calcium spike (see Fig. 8b of Stern andArmstrong, 1997). However, while Eq. (11) suggeststhat Ic can only be activated by very elevated calciumconcentrations (half-activation = 470 nM), our data(recall Fig. 4) indicates that such bulk concentrationsare only achieved after prolonged bouts of activity. Wetherefore propose that, since the calcium sensor foreach BK channel is typically less than 1 µm from acalcium channel (Storm, 1993), each BK channel isactivated by calcium in small region close to the chan-nel and not by bulk calcium (see also Chad and Eckert,1984; Dopico et al., 1999; Van Goor et al., 2001). Suchsmall domains can experience large, rapidly attenu-ating, calcium transients, and we model the dynam-ics of CBK with Eq. (8), with γ = BK. Since Ic isonly activated during strong depolarization, e.g. dur-ing a spike, the exact time course of CBK does not af-fect the spiking dynamics and so we assume that CBK

decays fully between spikes with a time constant of1 ms.

Repolarization of the action potential in other ver-tebrate preparations typically has a component carriedby Ic (Adams et al., 1982a; Storm, 1987; Shao et al.,1999; Warman et al., 1994). Our previous data (Sternand Armstrong, 1997) only concerned calcium spikesand furthermore Dopico et al. (1999) investigated dis-sociated cells, and so we re-examined the effect of Ic

blockade on spike repolarization with 100 nM IbTXin ten neurons using whole cell recording (Fig. 10).We find that blocking Ic slightly but significantly in-creases width at the base of spike by a mean differ-ence of 0.11 ms (p < 0.02), but has little effect athalf-amplitude (p < 0.13). No effects were observedon spike height or rise time. Therefore repolarizationis mediated by other currents (e.g. IA and IK ), and Ic

only becomes active toward the end of the spike, whichis compatible with the results of Dopico et al. (1999).

In contrast, block of Ic significantly decreases theHAP amplitude by ∼2 mV (p < 0.03) and hastens itsdecay by ∼13 ms (p < 0.006) (Fig. 10). We there-fore infer that the HAP has two components: the first iscaused by IA and IK overshooting rest and hyperpolar-izing the cell, but the after-potential is then enhancedand sustained by the late activation of Ic. The effects

378 Roper et al.

Figure 9. (a) Activation of AHP versus bulk calcium increase, both experiment (left) and model (right) stimulated at 40 Hz with 5 ms pulses.Note that spikes have been clipped for clarity. (b) Spike frequency adaptation: experiment shown above, model below. Note that both cell andmodel are held depolarized by ∼3 mV to fully inactivate IA . Experimental data taken from whole cell recordings from the SON in a hypothalamicslice.


Figure 10. The effect of pharmacological block of Ic with 100 nM IbTX, and simulation of block by setting Gc = 0 in the model. Left panelshows experiment, right panel shows model. Spike evoked with 3 ms depolarizing pulse and hence the initial ramp prior to spiking. Bold lineshows control, dashed line shows block of Ic . The HAP is still extant when Ic is blocked because IA and IK still overshoot the rest potential,but it decays faster and is reduced in amplitude. Experimental data taken from a whole cell recording from the SON in a hypothalamic slice.

of application of IbTX in vitro are compared with themodel in Fig. 10.

3.5. IK ,leak and the Depolarizing After-Potential

The DAP typically lasts for a few seconds, and tran-siently lifts the potential to ∼3 mV above rest, althoughits maximum amplitude depends strongly upon voltage(Bourque, 1986; Li and Hatton, 1997b). DAP’s there-fore briefly elevate the probability that incident exci-tation following the initial HAP will trigger a secondspike. If the cell does fire again then the DAP follow-ing the second spike sums with the residue of the firstto produce a much larger depolarization (Andrew andDudek, 1984a). Importantly, for AVP neurones DAPsummation appears to underlie a plateau potential thatdrives bursts of activity (Andrew and Dudek, 1983;Roper et al., 2001, 2003). The DAP depends stronglyon both voltage (Bourque, 1986; Li and Hatton, 1997b)and calcium (Andrew and Dudek, 1984a; Bourque,1986; Li et al., 1995). The DAP amplitude is maximumwhen the holding potential is ∼10 mV below thresh-old, and its amplitude decreases to zero as the holdingpotential is hyperpolarized below this level (see forexample Fig. 3 of Li and Hatton (1997b). This voltagedependence does not solely derive from the ohmic driv-ing force since DAP’s vanish at ∼ −70 mV rather thanat the K+ reversal potential (EK = −96 mV; Bourque(1988)). In addition: (i) chelation of [Ca2+]i ; (ii) block-ade of Ca2+ channels; (iii) blockade of calcium release

from stores; and (iv) removal of extra-cellular Ca2+ allinhibit the DAP (Li et al., 1995). In contrast: (i) asingle calcium spike can trigger a DAP when INa isblocked with TTX (Andrew and Dudek, 1984a); (iii)enhancement of calcium release from stores with caf-feine amplifies the DAP (Li and Hatton, 1997a); and(iii) application of antibodies to the calcium buffer cal-bindin can unveil a DAP in a neurone that would nototherwise express one (Li et al., 1995). Finally, oursimultaneous electrical recording and calcium fluores-cence shows that DAP evolution closely follows that ofintra-cellular calcium (recall Section 3.1 and Fig. 4).

A mechanism underlying the DAP is thought (Liand Hatton, 1997b) to be a Ca2+ -mediated modula-tion of a resting K+ current, possibly TASK-1 (North,2000; Brown, 2000), which produces a brief depolar-izing shift of the cell’s resting potential. A related,but distinct, mechanism has been proposed to underliethe depolarized after-potential in neocortical neurones(Greene et al., 1994), and the inhibition of a potas-sium leak underlies both the cooling-induced depolar-ization in thermally responsive dorsal root ganglioncells (Reid and Flonta, 2001) and the light-induced, per-sistent depolarization observed in Hermissenda type Bphotoreceptors (Blackwell, 2002). However note thatBourque et al. (Bourque, 1986; Ghamari-Langroudiand Bourque, 2002) have proposed that the DAP is in-stead driven by the activation of a non-specific cation(NSC) current rather the modulation of a resting potas-sium current. We consider this alternative mechanism,and its implications, in (Roper et al., 2003) and we have

380 Roper et al.

shown that the two models are formally equivalent pro-vided that the NSC current is partially activated at restand is modulated by calcium in a similar manner to thatdescribed above.

To model the DAP we first follow McCormick andHuguenard (1992) and subdivide Ileak into two linearcomponents: a sodium leak INa,leak (Lancaster, 1991)and a potassium leak IK ,leak, according to Eq. (7). How-ever we further allow IK ,leak to be modulated by bothCi and V . Since the DAP represents a calcium- andvoltage-dependent reduction in IK ,leak, we may writethis current as

IK ,leak = G K ,leak(1 − f ([Ca2+]i , V ))[V − EK ] (12)

and so

Ileak = GNa,leak[V − ENa] + G K ,leak(1 − f )[V − EK ]

(13)

(where ENa = 49.5 mV (Cobbett and Mason, 1987)and EK = −96 mV (Bourque, 1988)). The maximalconductances GNa,leak and G K ,leak may be determinedfrom: the resting membrane potential Vrest = −65 mV,the membrane time-constant τ = C/G leak = 12 ms(these measurements are an average of those measuredby Stern and Armstrong (1996)), and the boundary con-dition V = 0 at V = −65 mV. From which it fol-lows that GNa,leak = 0.018 mS cm−2 and G K ,leak =0.066 mS cm−2.

Although (1 − f (Ci , V )) describes the modulationof a K+ -leak conductance, it is simpler to think of f asbeing an activation function for the DAP. Neither theform nor the time-course of f is known, but as a firstapproximation we assume that its steady state value canbe written as

f∞(Ci , V ) = λ + �a∞(Ci )b∞(V ) (14)

where λ and � are constants, with λ + � = 1, thataccount for the possibility that IK ,leak is only partiallyinhibited. For simplicity we will assume that � = 1,λ = 0 here. The calcium- and voltage-dependent acti-vation functions are given by

a∞(Ci ) = tanh

(Ci − Cr

ka

)and

b∞(V ) =[

1 + exp

(− V + Vb

kb

)]−1

(15)

where ka = 50 nM, kb = 8 mV, Vb = −60 mV andthe resting calcium concentration is Cτ ≡ [Ca2+]rest =

Figure 11. Putative steady-state voltage- and calcium-dependentmodulation of the potassium leak current, IK ,leak (see Eq. (14)).

113 nM. We have chosen this functional form and pa-rameters to fit the experimental data for the voltage-and calcium-dependence of the DAP. However, up toa translation and scaling factor, our predicted calciumdependence (a∞(Ci )) is very close to that measuredby Selyanko and Sim (1998) for resting potassiumcurrents in cultured rat hippocampal pyramidal neu-rones. To account for the slow time-to-peak (∼0.3 s)we assume the calcium response to be slow but thevoltage response to be instantaneous, and so a obeysEq. (5) with τa = 75 ms. The DAP activation func-tion, f∞, is graphed in Fig. 11 and comparisons be-tween model DAP’s and experiment are shown inFig. 12.

3.6. ISOR and Rebound Firing

Oxytocin cells exhibit a sustained outward rectifica-tion (SOR) that becomes active when the cell is heldclose to threshold (Stern and Armstrong, 1995, 1996,1997). Underlying the SOR is a potassium current(ISOR) that is similar to the more common M-current(Adams et al., 1982b), but is insensitive to muscarine,and is also distinct (Stern and Armstrong, 1997)from the hyperpolarization-activated non-specificcation current, Ih (Ghamari-Langroudi and Bourque,2000). This current is not strongly expressed byAVP neurones, and so a test for its presence reliablydiscriminates cell types in vitro. The effect of ISOR isseen most clearly when the current is first activatedand then allowed to de-activate. This is done bycurrent-clamping the cell close to threshold, and thenapplying a lengthy hyperpolarizing pulse to below thecurrent threshold (Stern and Armstrong, 1995). ISOR


Figure 12. Experimental (left) and model (right) DAP’s and their associated calcium transient. Spikes are evoked by a 5 ms depolarizing pulseand have been truncated for clarity. Experimental data taken from whole cell recordings from the SON of a hypothalamic slice.

de-activates slowly and the membrane depolarizes asit does so, and so the signature of this current is anexponentially depolarizing sag to the voltage trajec-tory, as shown in Fig. 13a. (Note that the amplitudeof the depolarizing sag becomes diminished as thehyperpolarization approaches the potassium reversalpotential). Subsequent removal of the hyperpolarizingstimulus allows the membrane potential to recover.However because re-activation of ISOR is also slow,the voltage then transiently over-shoots the holdingpotential and gives rise to a brief, post-inhibitory,rebound depolarization (RD). The RD quickly decaysas ISOR reactivates, but can be of sufficient magnitudeto sustain a brief volley of action potentials.

When recorded in the presence of TTX, the RD ap-pears as a transient depolarization of the membrane. Itsmaximum amplitude follows a bell-shaped function ofthe amplitude of the preceding hyperpolarizing pulse(Stern and Armstrong, 1995), first increasing with hy-perpolarization until it reaches a maximum of ∼10 mVat a holding potential of ∼ −65 mV, and thereafterdecreasing. Application of 6 mM 4-AP significantlyenhances the RD and eliminates its decrease at hy-perpolarized potentials (Stern and Armstrong, 1997),indicating that reduction of the RD is due to removalof inactivation of IA. The model shows a similar de-pendency of the RD amplitude, and Fig. 13b demon-strates that the number of rebound-evoked spikes first

increases and then decreases for increasing hyperpolar-izations. Following the decrease, the number of evokedspikes in some cells will then saturate with increasinghyperpolarization, while in others it decreases to zero(data not shown). The amplitude of the RD in a particu-lar cell, and the corresponding number of spikes, likelydepends on the relative contributions of IA and ISOR.

This current has not yet been characterized involtage clamp, and so the voltage dependence and timeconstants of its activation have not been determined.However measurement of the RD amplitude fromincreasing hyperpolarizations in the presence of 6 mM4-AP (to block IA) shows that the current begins toactivate close to −65 mV and becomes fully activeat ∼−50 mV (see Fig. 7D of Stern and Armstrong(1997)). We have therefore assumed first-order kineticswith sigmoidal activation, and half-activation −60 mV.Activation and de-activation of ISOR appear to havedifferent time constants since the depolarizing sag canbe fit to a single exponential, τ = 0.39 s, while the RDdecays with time constant, τ1 = 0.26 s and τ2 = 0.21 s(Stern and Armstrong, 1995). However distinct timeconstants for activation and de-activation are not easilyincorporated into the Hodgkin-Huxley formalism andso for simplicity we have taken the activation timeconstant to be τSOR = 0.26 s, and note that this issignificantly slower than the activation of any othercurrent.

382 Roper et al.

Figure 13. Sustained outward rectification (SOR) and rebound depolarization (RD), right panel model, left panel experiment. (a) ISOR isactivated when V is held close to spike threshold and slowly de-activates when a hyperpolarizing pulse is applied—note consequent depolarizingsag to voltage trajectory. ISOR remains transiently de-activated when the hyperpolarizing pulse is released and the cell expresses a brief rebounddepolarization (RD) which can support a short spike train. (b) The magnitude of the RD follows a bell shaped function of the hyperpolarization,first increasing as de-activation becomes complete and then decreasing as IA de-inactivates. Experimental data is from a sharp electrode recordingfrom an OT cell in the SON of the hypothalamic explant. Note that all spikes have been clipped for clarity.

4. Discussion

We have developed mathematical models of bothspecies (OT and AVP) of SON magnocellular neurones.The models include many of the currents contributing

to the regulation of their firing patterns, and reproducea wide range of experimental protocols. While we haverelied largely on published reports of electrical proper-ties to parameterize our models, we have also presentednew pharmacological data showing that a significant


component of the HAP is sustained by Ic. In additionwe have presented new calcium imaging data that al-lows estimation of intracellular calcium transients con-trolling AHP’s, HAP’s and DAP’s in these neurones.

4.1. Calcium Buffering and Calcium Dynamics

Our main experimental finding is the dissociationof time-scales between the rise and the clearanceof somatic (bulk) calcium and the activation andde-activation of two of the three Ca2+ -activatedK+ channels that we have studied. We interpret thisdissociation as implying strong calcium compartmen-talization within the cell, which is possibly caused bycalcium buffers. In fact Ca2+ buffering is known to bea crucial determinant of firing activity in these cells (Liet al., 1995). For example, the intrinsic buffers calre-tinin and calbindin-D28K are localized to OT but notAVP neurones of the SON (Arai et al., 1996, 1999;Miyata et al., 1998). Recall that a key distinction be-tween OT and AVP cells is the preferential ability ofthe latter to express a DAP. The presence or absence ofthese buffers modulates the DAP expression, since in-jection of calbindin-D28K anti-serum unmasks the DAPand converts regularly firing cells to phasic cells. Theconverse is also true: injection of calbindin-D28K intophasic cells suppresses the DAP and changes activityfrom phasic to continuous firing (Li et al., 1995).

4.2. Inhibitory Currents Regulating Excitability

In the normal rat, both OT and AVP cells emit a Poissondistributed slow-irregular ∼1.5 Hz spike-train (Poulainet al., 1988). Each spike is caused by synaptically-driven random threshold crossings, and such firing pat-terns have been termed “occasional spiking” (Calvin,1975). This discharge is tightly regulated by the in-hibitory K+ currents, Ic, IAHP, IA and ISOR, and in factMNC’s will actively defend their intrinsic firing rateand compensate for evoked spikes by slowing theirspontaneous discharge (Leng et al., 1995).

Our model shows that while each current plays aunique role in regulating firing, there are duplicatemechanisms for both short- and long-term inhibition inthese cells that can be activated either by spike activity(Ic and IAHP) or by modulations of the rest potential(IA and ISOR).

First Ic effects the post-spike refractory period viathe HAP (Fig. 10), and so transiently lessens the cell’s

response to synaptic input. The function of the AHP issimilar but it acts over a more sustained period (Fig. 9),and is only activated when two or more spikes are emit-ted proximally. Both of these currents therefore regu-late discharge as a function of the cell’s spiking historyand maintain the variability of the spike train (Cazaliset al., 1985) by ensuring that short interspike-intervalstend to be followed by much longer intervals.

In contrast, the voltage-dependent currents IA andISOR do not depend on prior activity but instead con-trol excitability via the membrane potential. They aretherefore more responsive to extraneous factors thatmanipulate the resting potential, such as synaptic inputor neuromodulation, rather than intrinsic spike activity.The relative time-courses of IA and ISOR complementthose of Ic and IAHP: IA recovers swiftly from inac-tivation and inactivates progressively during rapid fir-ing, it therefore mediates short-term inhibition. On theother hand, ISOR is slow and sustained and induces along-term inhibition. Furthermore, these two currentsare modulated by opposing stimuli: the removal of A-current inactivation (Fig. 7) is aggravated by longerhyperpolarizations, while ISOR is activated by steadydepolarizations (Fig. 13). Thus the excitability of OTcells (those that express ISOR) is a bell shaped functionof the holding potential (recall Fig. 13) while that ofAVP cells (which do not express ISOR) is an increasingfunction of the holding potential.

All four of these currents are targets for neuromod-ulators and can be up- or down-regulated during dif-ferent physiological stresses. For example, during bothpregnancy and lactation, the response of OT cells toother OT-releasing stimuli such as hyper-osmolality(Higuchi et al., 1988) is blunted. Coincidentally, theAHP (Teruyama and Armstrong, 2002) is up-regulatedduring this time and is likely to contribute to this di-minished response. However, the decay of the AHP isalso faster in OT neurons during lactation despite thisincreased amplitude, suggesting a complex scenario ofstate dependent AHP modulation. In addition, lactationinduces an increase in the amplitude of the rebound-depolarization (RD) (Stern and Armstrong, 1996), butleaves the apparent SOR amplitude unaffected, sug-gesting a modulation of τSOR rather than the conduc-tance. Alternatively, noting the strong interaction be-tween IA and the rebound depolarization, this could becaused by a change in the properties of IA. All fourcurrents can also be regulated indirectly, and can betempered by activation of metabotropic glutamate re-ceptors (mGluRs) (Schrader and Tasker, 1997). mGluR

384 Roper et al.

up-regulation occurs during dehydration (Meeker et al.,1994) and probably further promotes cellular excitabil-ity during phasic firing.

4.3. The Osmotic Response

Although both hormones are released during hyper-osmolality or dehydration, the two cell types responddifferently to an increasing stress (Wakerley et al.,1978; Roper and Sherman, 2002). OT cells switch to afast-continuous discharge that rarely exceeds 10 Hz,while AVP cells fire a phasic discharge pattern oflengthy repeating bursts, each of which rides upon aplateau potential (Andrew and Dudek, 1983). In Roperet al. (2003) we have shown that our model can alsoaccount for the plateau potential and for phasic activity.

The osmo-response in both cell types is mediatedby a sustained depolarization of up to +15 mV, whichhas several contributing factors (see e.g. Voisin andBourque, 2002). In our model for AVP cells, this net de-polarization traverses the range of the voltage sensitiv-ity of the DAP (Fig. 11), while for OT cells it traversesthe activation of ISOR (Fig. 13).

The plateau potential in phasic AVP cells is abovespike threshold and is sustained by the regenerativesummation of DAP’s (see Fig. 14a). Thus, since de-hydration increases DAP amplitude, it also facili-tates phasic activity. However, although ∼30% of OTcells express a DAP (Armstrong et al., 1994), theyrarely discharge phasically (Brimble and Dyball, 1977;Armstrong et al., 1994). Our model indicates that thisdisparity is due to the activation of ISOR as the mem-brane potential approaches spike threshold. Therefore,although OT cells also exhibit DAP summation whenconcurrent spikes are evoked, ISOR inhibits the celland prevents it from spiking regeneratively. SummedDAP’s then decay with their usual time-course (seeFig. 14c).

4.4. A Dual Role for Calcium

By activating the AHP, calcium is usually perceived as amessenger of electrical inhibition rather than excitation(see e.g. Vergara et al., 1998). However in AVP cells itplays two opposing roles: not only does it mediate bothshort- and long-term inhibition via the activation of BKand SK channels (Ic and IAHP), but it can also excite thecell by turning off the resting K+ current, IK ,leak (Li andHatton, 1997b). Having two such antithetical functions

Figure 14. DAP summation and the initiation of phasic activity. (a)sharp electrode recording from SON of hypothalamic explant, (b)SON model with DAP, (c) SON model with DAP and SOR. (a) and(b): Three spikes are evoked by a short depolarizing pulse, and thesubsequent DAP summation carries the membrane potential abovespike threshold and causes a regenerative plateau potential. (c) Thesame model and protocol as (b) but with ISOR included. The SORinhibits the plateau and prevents regenerative firing, and the summedDAP decays to rest with the usual slow time course.


appears at first glance to be contradictory. Howeverthere is a delicate interplay between the DAP and theAHP that is dictated by their respective time-scales,and whether the cell exhibits a depolarizing or a hy-perpolarizing after-potential depends both on the stim-ulation frequency and also the number of spikes in thetrain (Andrew and Dudek, 1984b). In phasic neuronsstrongly expressing DAP’s, prolonged low frequencystimulation (<1 Hz) evokes single spikes and conse-quent DAP’s, which then fade to the resting poten-tial. At intermediate frequencies (>1 and <10 Hz) theDAP’s sum to form a regenerative depolarized plateaupotential which can then support endogenous firing viapositive feedback (Andrew and Dudek, 1983). In con-trast, at higher frequencies (>10 Hz) the AHP becomespredominant and instead a net hyperpolarization occurs(see Fig. 1b). Thus calcium acts here both to amplifysignals of a certain bandwidth, and to filter stimulationthat is of too high or too low a frequency. Furthermore,since the DAP is also voltage-dependent, this signalprocessing is labile and depends on the membrane po-tential. Thus dehydration magnifies the gain of the am-plification, and contributes to the phasic activity seenin AVP cells under increased osmotic stress (Wakerleyet al., 1978).

Appendix

The full model is given by

dV

dt= − 1

C (INa + ICa + IK + IA + Ic

+ IAHP + ISOR + Ileak)

and we have assumed the specific capacitance to beC = 1.

Recall from Eq. (4) that inactivating and non-inactivating currents are respectively described by

Iγ = gγ mαhβ(V − Eγ ) or Iγ = gγ mα(V − Eγ )

and their corresponding voltage-dependent activation(m∞) and inactivation (h∞) functions are

m∞(V ) =(

1 + exp

(−V − Vm

km

))−1

and

h∞(V ) =(

1 + exp

(V + Vh

kh

))−1

Reversal potentials are ENa = 50 mV, EK =−96 mV, ECa = 12.5 ln([Ca2+]o /[Ca2+]i ), and wehave assumed the external calcium concentration to be[Ca2+]o = 4 mM.

The parameters for each voltage-dependent currentare tabulated below

gγ Vm km Vh kh

(mS cm−2) (mV) (mV) (mV) (mV)

INa = gNam3∞h(V − ENa) 14 38 4 45 2

ICa = gCam2∞(V − ECa) 0.1 −10 7 – –

IDR = gDRm3(V − EK ) 14 −2 11 – –

IA = gAm4h(V − EK ) 14 45 11 80 6.5

ISOR = gSORm(V − EK ) 0.06 60 1.8 – –

IAHP = gAHPq2∞(V − EK ) 0.18 – – – –

Ic = gc p(V − EK ) 1 – – – –

and the corresponding time constants (in ms) for eachcurrent are

INa τm –τh

20(1+exp

(V −1

14

))(1+exp

(V +30

34

)) − 0.075

ICa τm

(0.19

(−V +19.88

exp( −V +19.88

10 −1))

+ 0.046 exp( −V

20.73

))−1

τh –

IDR τm8

exp(

V +4019

)+exp

( −V −4020

) + 1.8

τh –

IA τm35

1.1 exp(

V +4317

)+exp

( −V −2013

)τh 5 + 120

exp(

V +7530

)+exp

( −V −4518

)IAHP τm –

τh –

Ic τm 1.22

τh –

ISOR τm 260

τh –

The calcium-dependent activation functions for IAHP

and Ic (respectively q and p) have been describedpreviously in the text, but for completeness theyare

q∞(CSK)

=(

1 + exp

[−1.120 − 2.508 log

(CSK − Cr

1000

)])−1

386 Roper et al.

p∞(CBK, V ) =[

1 + 470

C2.38BK

]−1

×[

1 + exp

((−V − 140 log10 CBK + 370)

7.4

)]−1

where CBK denotes the calcium concentration in thepool sensed by the BK channel, CBK denotes theconcentration in the AHP calcium domain, and Cr

(= 113 nM) the resting calcium concentration.Each of the three calcium pools evolves according

to (recall Eq. (8))

d

dtCγ = −αγ ICa − 1

τγ

(Cγ − Cr )

with time constants τi = 2.33 s (Ci ), τBK = 1 ms (CBK)and τSK = 656 ms (CSK), and Faraday scaling factorsαi = 1.4, αBK = 100 and αSK = 1.6.

The leak current is the linear sum of two distinctcurrents

Ileak = INa,leak + IK ,leak

and IK ,leak can be modulated by both intra-cellularcalcium and voltage according to

f∞(Ci , V ) = λ − �a∞(Ci )b∞(V ) (16)

where

a∞(Ci ) = tanh

(Ci − Cr

ka

)and

b∞(V ) =[

1 + exp

(− V + Vb

kb

)]−1

(17)

with ka = 50 nM, kb = 8 mV, Vb = −60 mV, theresting calcium concentration is Cr = 113 nM, andwhile b activates instantaneously, a evolves with τa =75 ms.

Acknowledgments

We would like to thank Arthur Sherman for guidanceand support during the preparation of this work, andBard Ermentrout for releasing, and supporting XP-Paut (http://www.math.pitt.edu/∼bard/xpp/xpp.html), whichwas used for all of our numerical work. We would alsolike to thank David Brown, Gareth Leng and FrancoiseMoos for advice on an earlier version of this model,

and Victor Matveev for a careful reading and critiqueof this manuscript.

This work was supported by an NIH IRTA Fellow-ship (PR) and by NIH grant numbers NS23941(WEA)and NS42276(JCC).

Note

1. In other preparations this after-potential is often termed the fastAHP (fAHP) (see e.g. Storm, 1987), but the term HAP is usualin the SON MNC literature (Bourque et al., 1985).

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